This formalization is often understood as being the default meaning of “quantum logic”. But the proposal has later been much criticised, for its lack of key properties that one would demand of a formal logic.

The term quantum logic is usually understood in connection with the 1936 Birkhoff-von Neumann proposal to consider the (closed) linear subspaces of a Hilbert space ordered by inclusion as the formal expression of the logical distinction between quantum and classical physics. While in classical logic we have deduction, the linear subspaces of a Hilbert space form a non-distributive lattice and hence there is no ob vious notion of implication or deduction. Quantum logic was therefore always seen as logically very weak, or even as a non-logic. In addition, it has never given a satisfactory account of compound systems and entanglement.

Attractive and revolutionary as this spatial quantum “logic” may appear it faces severe problems. The main logical drawbacks are:

Due to its lack of distributivity, quantum ‘logic’ is difficult to interpret as a logical structure.

In particular, despite various proposals no satisfactory implication operator has been found (so that there is no deductive system in quantum logic).

Quantum ‘logic’ is a propositional language; no satisfactory generalization to predicate logic has been found.

Quantum logic is also problematic from a physical perspective. Since (by various theorems and wide agreement) quantum probabilities do not admit an ignorance interpretation, [0,1][0, 1]-valued truth values attributed to propositions by pure states via the Born rule cannot be regarded as sharp (i.e. {0, 1}-valued) truth values muddled by human ignorance. This implies that, if X=[a∈Δ]X = [a \in \Delta] represents a quantum-mechanical proposition, it is wrong to say that either xx or its negation holds, but we just do not know which of these alternatives applies. However, in quantum logic one has the law of the excluded middle in the form x∨x⊥=1x \vee x^\perp = 1 for all xx. Thus the formalism of quantum logic does not match the probabilistic structure of quantum theory responsible for its empirical content.

But notice that one may argue that the first three points here are squarely resolved by thinking of BvN-quantum logic as embedded into linear logic, we come back to this in a moment. Concerning the last point one might argue that the propositions in BvN-quantum logic concern the quantum measurement-outcomes (only), for which, at least in some interpretations, it does make sense to speak of a definite result.

Among the magisterial mistakes of logic, one will first mention quantum logic, whose ridiculousness can only be ascribed to a feeling of superiority of the language – and ideas, even bad, as soon as they take a written form – over the physical world. Quantum logic is indeed a sort of punishment inflicted on nature, guilty of not yielding to the prejudices of logicians… just like Xerxes had the Hellespont – which had destroyed a boat bridge – whipped.

both orthologic (or weakenings thereof) and linear logic share the failure of lattice distributivity. In particular, the fragment of linear logic that includes just negation and the additive connectives is nothing but a version of the paraconsistent quantum logic PQL,

(Girard 87 introduces linear logic nad suggests a possible relation to quantum physics, but remains undecided on thatM on p. 7 it says: “One of the wild hopes that this suggests is the possibility of a direct connection with quantum mechanics… but let’s not dream too much!”)